Set 6 Problem number 9

• Problem

• Solution

Problem

Consider the region between the horizontal axis and the points ( 9 , 5 ) and ( 12 , 19 ) on a graph of the net force parallel to the x axis on an object, in Newtons, vs position in meters along the x axis?

• What is the area of this region? 
• What is the meaning of this area?

Solution

The area under the segment will consist of a trapezoid with altitudes 19 Newtons and 5 Newtons, and uniform width ( 12 meters - 9 meters) = 3 meters.

• The average of the altitudes represents the average of the forces 19 Newtons and 5 Newtons.

• If force is changing at a uniform rate the segment in fact represents the force vs. position precisely; otherwise it is only an approximation to the behavior of a curving graph.
• In general we therefore say that the average of the two forces is the approximate, not the exact, average force on the interval.

• The width of the trapezoid represents the displacement over which this approximate average force is sustained.
• The area of the trapezoid is the product of the average altitude, representing the approximate average force on the interval, and the width, representing the displacement.

• The product of net force and time interval is equal to the work done on the object, which will be the change in the KE of the object.
• The area therefore represents approximate average force * displacement = work done on object = change in KE of the object during the time interval.

The average altitude in the present example is ( 5 Newtons + 19 Newtons) / 2 = 12 Newtons.

• The area is thus 12 Newtons * 3 meters = 36 Newton meters = 36 Joules
• We interpret this as the approximate work done on the object, which is the approximate change in KE corresponding to the two points on the graph.